Algebra 2 Semester 1 (#2221)

Transcription

1 Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Course Guides for the following course: Algebra 2 Semester 1 (#2221) When used as test practice, success on the Instructional Materials does not guarantee success on the district math common final. Students can use these Instructional Materials to become familiar with the format and language used on the district common finals. Familiarity with standards and vocabulary as well as interaction with the types of problems included in the Instructional Materials can result in less anxiety on the part of the students. The length of the actual final exam may differ in length from the Instructional Materials. Teachers can use the Instructional Materials in conjunction with the course guides to ensure that instruction and content is aligned with what will be assessed. The Instructional Materials are not representative of the depth or full range of learning that should occur in the classroom. *Students will be allowed to use a Scientific or graphing calculator on Algebra 2 Semester 1 and Algebra 2 Semester 2 final exams. Page 1 of 42

2 Algebra 2 Semester 1 Test Reference Sheet Sum of Two Cubes Difference of Two Cubes Note from Earl. Many of the solutions in this document use techniques presented in the Algebra Handbook, which is available on the website. If you have trouble following any of the techniques used, try looking in the handbook for pages that deal with the issue you are struggling with. I solve the problems in this test using the quickest method available in most cases. Occasionally, I also make comments about some of the math involved in an effort to enhance your understanding of what is going on in the problem. Multiple Choice: Identify the choice that best completes the statement or answers the question. Page 2 of 42

3 1. Solve the following equation for y: A. 42 C. 5 2 B. 412 D. 5 2 To solve an equation in terms of a designated variable, perform the following steps: Gather all terms containing the variable on one side of the equal sign, and all terms not containing the variable on the other side of the equal sign. Factor out the variable in question. Divide both sides of the equation by the factor that does not contain the variable. Here we go: Original equation: Subtract 7 from both sides: 7 7 Result: Factor out : Divide by 2 : 2 2 Result: Answer D Page 3 of 42

4 2. Which graph represents the piecewise function? 1 1, , 3 3 A. C. B. D. First, note that the breakpoint for the function occurs at 3. In terms of the above graphs, this leaves out A (which has a gap) and D (which is not a function). Next, notice that following about : The piece of with a positive slope (since 1) is the piece to the right of 3. Of the remaining answers, B and C, this occurs in C and not in B. Answer C Page 4 of 42

5 3. Which of the following piecewise functions represents the graph below? 3 1, 2 A , 2 1 2, 1 3 1, B , 2 1 2, 1 3 1, C , 2 1 2, 1 3 1, 2 D , 2 1 2, 1 Left Center Right Piece Piece Piece Let s look carefully at the pieces in the graph. First, note that the breakpoints for the function occur at 2 and 1. The solid dots in the graph occur at the left endpoints of both the middle piece and the right piece. Solid dots correspond to " " or " " signs in the notation given. This leaves out C and D, both of which have < signs on the left side of the middle piece. Next, notice that the slopes on the left and middle pieces in the graph are positive and the slope of the right piece is negative. Of, the remaining answers, A and B, this occurs in A and not in B. Answer A Page 5 of 42

6 4. Which of the following is a proper description of the range of the function shown: 1 2 A. : 36 C. : 21 B. : 36 D. : 21 Recall the following definitions: The Domain of a function is the set of all values of for which it is possible to calculate a value. The Range is the set of values that function takes on for all in the function s Domain. To show the range, I like to locate a function s bottom and/or top values and draw a line between them off to the side of the graph. This in indicated by the lines to the left of the figure above. (Note: if the range is unlimited in either the up or down direction, I would show an arrow in that direction instead of a line.) Notice that I drew a solid magenta line for 1, indicating that there is a closed dot at the point 6, 1. I drew a dashed magenta line for 2, indicating that there is an open dot at the point 3, 2. Based on this, the range is the set of all real numbers from 2 to 1, excluding 2 because of the open dot at the point 3, 2. In set notation, the range would be shown as: : In interval notation, the range would be shown as:, Answer D Page 6 of 42

7 5. Which of the following is the graph of over the domains 3, 3 5, 7? A. C. B. D. Recall that the parent absolute value function is:, where, is the vertex. What do we know about the equation given: for 3, 3 5, 7? It has vertex:,, or, in mixed numbers: 1,3. All of the solutions have this property. No help for us here. It forms a V shape with a sharp point because it is an absolute value function. This eliminates C. It opens upside down because of the minus sign in front of the absolute value sign. This eliminates D. It does not have any values between 3 and 5 because must be in one of the intervals 3, 3 or 5, 7. All of the solutions have this property. No help for us here. It has an open point at 3 because of the parenthesis in 3, 3. It has a closed point at 5 because of the bracket on the left of 5, 7. This eliminates A. The only solution left is B, and it fits everything we know about the function. Answer B Page 7 of 42

8 6. Three students were chosen to show their solutions for solving the equation for x. Their work is shown below. Determine which students were correct. Student #1 Student #2 Student #3 A. Student #1 and Student #2 B. Student #2 and Student #3 C. Student #1 and Student #3 D. All students were correct Student 1 is correct. Student 2 forgot to divide the constant by in step 2. Student 3 is correct. (Note that the solution shown is equivalent to that shown for Student 1.) Answer C Page 8 of 42

9 7. Solve the following system for y: A. 15 B. 4.6 C. 3 D. 2.5 These can be a little tricky, so let s look for a good way to start. Notice that the first two equations have the same and terms. Let s eliminate and to find the value of. We can do this by multiplying the second equation by 1 and adding it to the first equation: First: 3 Second 1: 5 Next, let s re write the 2 nd and 3 rd equations, making 1. Note that, alternatively, we could use the first and 3 rd equations. We just can t use the first two, because we already used that pair to find. We must use a different pairing of the original equations. Equation 2: 5 becomes: 1 5, or, by subtracting 1: 4 Equation 3: 232 becomes: 1 232, or, by adding 1: Let s work with the two new equations to find the value of, as required by the problem. To obtain, we need to eliminate. So let s multiply the Revised Equation 2 by 3 and add it to the Revised Equation so so Answer C Page 9 of 42

10 8. Which equation is obtained after the translation of the graph up 2 units and left 6 units? A. 2 B. 2 C. 2 D. 2 The general form of an absolute value function is:, where, is the vertex of the function. The function shown has a vertex at 4, 2, and a slope of 1, so it s form would be: 4 2. Now, let s translate it (see the arrows in the above chart). Translation left 6 units means subtract 6 from. So, our new 4 6. Translation up 2 units means add 2 to. So, our new 2 2. These translations generate the equation, or Answer A Page 10 of 42

11 9. Which of the following is the graph of over the domain 3, 3 A. C. B. D. Recall that the parent absolute value function is:, where, is the vertex. What do we know about the equation given: over 3, 3? It has vertex:,, or, in mixed numbers: 1,2. All of the solutions have this property. It forms a V shape with a sharp point because it is an absolute value function. This eliminates A and C. It opens upside down because of the minus sign in front of the absolute value sign. This eliminates D. The only solution left is B, and it fits everything we know about the function. Answer B Page 11 of 42

12 10. The graph of is vertically compressed by a factor of and translated to the right three units and down one unit to produce the function. Which of the following equations represents? A C B D Recall that the parent quadratic function is:, where, is the vertex, is the stretch or compression, and the sign of determines whether the function opens up or down. Let s look at what we are told: There is no mention of a reflection, so is positive. We have a compression factor of. So,. The parent function is translated to the right three units and down one unit, so, 3, 1. The resulting equation, then, is: Page 12 of 42

13 11. Compare the two functions represented below. Determine which of the following statements is true. Function Function 6 4 This function: Opens up, Has vertex at (6, 4). Is not stretched or compressed. Is not reflected. A. The functions have the same vertex. Recall that the parent quadratic function is:, where, is the vertex. So, the vertex of the right function is 6, 4. The vertex of the left function is 2, 4 from the graph. So, this statement is FALSE. B. The minimum value of is the same as the minimum value of. Both functions open up, so the minimum value of each is the coordinate of its vertex (i.e., ). For the left function, this is 4 (from the graph), and for the right function, this is also 4. So, this statement is TRUE. C. The functions have the same axis of symmetry. The equation for the axis of symmetry is, where h is the coordinate of the vertex. For the left function, the axis is 2, and for the right function, the axis is 6. So, this statement is FALSE. D. The minimum value of is less than the minimum value of. In B, above, we determined that the minimum values were the same, so this statement is FALSE. Answer B Alternatively, graph on a calculator and read the required values off the graphs. Page 13 of 42

14 12. If the function is translated left eight units and up ten units, how will the domain and range of the function change? A. The domain will become : 8 and the range will become : 10. B. The domain will become : 8 and the range will become : 10. C. The domain will become : 8 and the range will remain :. D. The domain will remain : and the range will remain :. The Domain and Range of all polynomials with odd lead coefficients are All Real Numbers. Since this function has degree 3, which is odd, translated is irrelevant; the Domain and Range remain All Real Numbers. Answer D 13. Which of the following functions has a range of all real numbers? I II III IV Linear function: always has range:. Absolute value function: has a limited range because it has a vertex. Quadratic function: has a limited range because it has a vertex. Cubic function: always has range:. A. I only C. I and IV B. II only D. II and III Based on the above comments in blue, only and have unlimited ranges, that is, ranges that are all real numbers. Answer C Page 14 of 42

15 14. The function is graphed to the right. Over which intervals of x is the graph positive? A. B. C. D. Easy Peezy. Above the axis is the green part in the figure above. Where does the curve exist in this green area? Answer: between 3 and 0, as well as to the right of 2.5. On the number line, all intervals should be open. Answer A Page 15 of 42

16 Simplify: A. 22 B. 23 C D Answer A Page 16 of 42

17 You will need the quadratic formula for the next couple of questions: 4 2 provides solutions to the equation: What are the solutions to the quadratic equation, ? A. B C D First, combine all terms on one side of the equation. Original Equation: Subtract 5 7: Result: Next, use the quadratic formula with 3, 2, Answer C 21. What are the solutions to the quadratic equation, 295? A B First, combine all terms on one side of the equation. C D Original Equation: Subtract 5 9: Result: Next, use the quadratic formula with 1, 3, Page 17 of 42

18 22. Solve the following equation: 5 8 A. 23 C B. 43 D Original Equation: 5 8 Multiply by 6 to get rid of the fraction: 6 6 Result: Take square roots: Simplify: Subtract 5: 5 5 Result: Answer D 23. The function is graphed below. What are the solutions to 0? A. 8,0 B. 3,5 C. 42 D. 4 Real solutions are located where the curve crosses the axis. Since this function does not cross the x axis, it has no real solutions, which eliminates answers A and B. Important point you may not know: The real portion of complex solutions is, the coordinate of the vertex. So the complex solutions have the form: something. Since this equation has complex solutions and 4, those solutions are of the form: 4 something. Answer C Page 18 of 42

19 24. Given the function, 4 3, determine which of the following statements is true. A. The maximum value of is 3 and 0 has no real solutions. B. The maximum value of is 3 and 0 has two real solutions. C. The maximum value of is 4 and 0 has no real solutions. D. The maximum value of is 4 and 0 has two real solutions. Recall that the parent quadratic function is:, where, is the vertex. The maximum (or minimum) of a quadratic function occurs at its vertex, and is the coordinate of its vertex (i.e., ). This function opens down because of the negative sign in front of the quadratic term. Therefore, it has a maximum value. Since 3, the maximum value is 3. Since the maximum is a negative value, this function never crosses the axis and so it has no real solutions. Answer A Alternatively, graph the function on a calculator and read the required information off the graph. 25. Given the function, 27 find the minimum value. A. Minimum value is 5 B. Minimum value is 6 C. Minimum value is 7 D. Minimum value is 8 This function opens up because of the positive sign in front of the quadratic term. Therefore, it has a minimum value. The vertex of a quadratic in form occurs at: Substitute this into the original equation to find the minimum value of the function: Answer B Alternatively, graph the function on a calculator and read the required information off the graph. Page 19 of 42

20 26. Which of the following functions is equivalent to 611? A C B D Looks like we need to complete the square to put in, form Which function is represented by the graph? A. 2 3 B. 2 3 C. 2 3 D. 2 3 The zeros of the function are 2 and 3. The signs on zeros reverse when in factor form, so the factors are and. Answer C Page 20 of 42

21 28. Which of following functions does not represent the parabola with a vertex at 1, 4 and x-intercepts 1, 0 and 3, 0. A. 4 C. 23 B. 1 4 D. 1 3 Best way to approach this one may be with the answer that obviously has a vertex at 1, 4 answer B. So, answer B is not what we want. Let s try multiplying it out and then factoring to get the other two answers represent the same curve , so answer C is not what we want. Now, factor: , so answer D is not what we want. That leaves answer A as our solution. Answer A Alternatively, graph each function on a calculator to see if it has the required vertex and x intercepts. 29. Which of the following functions does not have zeros of 4 and 8? A C. 432 B D. 48 A zero of a function gives a value of zero when substituted into the equation. We could substitute 4 and 8 into each equation to see if we get 0, but that would take a lot of time. Let s try something else. Inspecting the answers, we see that answer D is in intercept form, and that its zeros are 4 and 8. Since we want the equation that does not have 4 and 8 as zeros, we have found it by inspection. Whew! That was a close one! Answer D Alternatively, graph each function and check its zeros (i.e., where it crosses the axis). Page 21 of 42

22 30. If 3 9, then what is the value of? A. 36 C. 18 B. 27 D. 9 We could multiply 3 9 out and complete the square, but there is a simpler approach. If you like this approach, use it, if not, multiply and complete the square (or graph as indicated in the alternative below). Here are the steps: We know that is the coordinate of the vertex. We also know that the coordinate of the vertex is half way between the zeros of the quadratic function. The zeros are obviously at 3, 9. Halfway between these is 3. Substitute 3 for in the original expression to get the value of : Answer A Alternatively, graph the function on a calculator and read the value of off the graph. Page 22 of 42

23 31. Compare the axis of symmetry and the minimum values for the two functions below Determine which of the following statements is correct. A. The functions and have the same axis of symmetry, but the minimum value of is less than the minimum value of. B. The functions and have the same axis of symmetry, but the minimum value of is greater than the minimum value of. C. The functions and do not have the same axis of symmetry, and the minimum value of is less than the minimum value of. D. The functions and do not have the same axis of symmetry, and the minimum value of is greater than the minimum value of. Axis of Symmetry We know that the axis of symmetry is half way between the zeros of a quadratic function. For, the zeros are 3, 7, so the axis of symmetry is 2. The vertex of a quadratic in form occurs at. So, for, the axis of symmetry is 2. Therefore, the functions have the same axis of symmetry. This eliminates answers C and D. Minimum Values Minimum values are determined at the vertex, and the vertex lies on the axis of symmetry. So, for both functions, the minima will be found at the function values where Therefore, the minimum value of is less than the minimum value of. Answer A Alternatively, graph both functions on a calculator and compare the axes of symmetry and minimum values of the functions from their graphs. Page 23 of 42

24 32. Compare the functions, and, and explain how the graph of 44 is related to the graph of 42. A. is vertically stretched to make B. is translated 6 units left to make C. is translated down 6 units to make D. is compressed vertically to make After the last couple of problems, it s good to get an easy one. Let s look at the two equations: They are pretty similar. The only difference between the two equations is that has a constant term that is 6 less than. Since the constant term indicates the vertical placement of the curve, is vertically 6 units below. Therefore, to get from, you would translate down 6 units. Answer C Page 24 of 42

25 33. Translate 46 five units to the left. What is the graph obtained after the translation? A. C. B. D. Notice that the vertices in the graphs are at four different locations. So, if we can find the vertex after the translation, we have our answer. The easiest way to determine the vertex of this function is to complete the square: which has vertex, 2, 2 To get the missing constant term, divide the coefficient of by 2, then square the result. In this case, Finally, we need to translate the vertex 5 units to the left. Our new vertex becomes: 2 5, 2, Answer C Page 25 of 42

26 34. A company is planning on selling plastic widgets. The company has to pay $600 in production costs plus $50 per widget to produce widgets. The equation, , models the total production costs. Based on sales of a similar product, the company expects their revenue to be modeled by 400. Use the equation to find the maximum profit,, the company can expect to make. A. $5,325 C. $15,025 B. $10,650 D. $40, The maximum of a quadratic function (with a negative lead coefficient) will always occur at the vertex. For an equation in form, this occurs at. For the above equation, 150. Finally, we need the coordinate of the vertex: ,650 Answer B Page 26 of 42

27 35. A local business wants to expand the size of their rectangular parking lot that currently measures 120 by 200. The project will cost less if equal amounts are added to each side, as shown below. Zoning restrictions limit the total size of the parking lot to 35,000. What is the maximum amount of distance that can be added on to the each side of the parking lot? Round your answer to the nearest foot. A. 1 C. 105 B. 31 D. 187 The size of the new parking lot will be: , which we want to be 35,000 square feet. This boils down to solving the equation: ,000. Original equation: ,000 Expand the expression: ,000 35,000 Subtract 35,000: 35,000 35,000 Result: ,000 0 Use the quadratic formula: Result: 351.3, 31.3 Only the positive solution makes sense in the context of this problem. Answer B Page 27 of 42

28 36. A parabola has a vertex of 5, 6 and passes through the point 10, 4. In the form of the parabola, what is the value of? A. 2 C B. D We start with the form:, where, is the vertex. The vertex of the left function is 5, 6, so our equation becomes: 5 6. To find the value of, we must substitute the values of 10 and 4 (from our point 10, 4) into this equation, and solve for. Original equation: Substitute values for and : Result: Subtract 6: 6 6 Result: Divide by 25: Result: Answer C 37. Which of the following systems of equations could a student use to write a quadratic function in standard form for the parabola passing through the points 1, 3, 4, 6, and 5, 9? 3 A B C D All of the solutions have three terms on the left, suggesting the use of the form:. Creating a system of equations like this requires us to replace the values of and in that form with the and values of the points given. So, we are looking for equations that do not have any s or s left in them. This eliminates answers A and C. The multipliers of in the equations are the values of from the points, so we are looking for coefficients of equal to 1 1, 4 16, and In the remaining answers, B and D, only D has these coefficients. Answer D Page 28 of 42

29 /3 FALSE Answer C 38. A parabola has x-intercepts at 2 and 6 and goes through the point 8, 4. What other point is on the parabola? A. 2, 2 C. 3, 15 B. 4, 2 D. 1, 15 Knowing the zeros of a quadratic function tells us a lot about its equation. With zeros of 2 and 6, we know that the equation has the form: 26, where is yet to be determined. Using the point on the line, 8, 4, we can determine the value of by substituting 8 and 4 into the equation and solving for : And, so our equation becomes: 2 6 To determine which point lies on the curve, substitute and of each point into this equation to see which ones make the equation work (1) : A. 2, B. 4, 2) FALSE FALSE C. 3, 15) TRUE D. 1, Note from Earl: We would stop working on this problem after finding the true solution in answer C. Unless I am missing something, this is a very time consuming problem and should be left until after all other questions have been answered. (1) Alternatively, graph the equation 2 6 and see which point is on the curve. Page 29 of 42

30 39. Solve the following system for the y-coordinates of the system: A C. 513 B. 3 5 D. 319 To get the values, we need the values, so let s just solve the system. Both equations are in the form " ", so let s set them equal to each other and solve for. Set the equations equal: Add 4 7: Result: Factor: Solve for : 5,3 Next, we substitute the values into either of the original equations to obtain the values. 5: 3: Answer A 40. What are the x-coordinates of the solution for the system given below? A. 2 4 C. 33 B. 4 2 D. 3 We want to eliminate. Notice that the term in the first equation is double the term in the second equation, so let s multiply the second equation by 2 and add it to the first Add 1 to both sides: 1 1 Result: Factor: Solve: Answer D Page 30 of 42

31 41. Given 3 65 and 323, complete the tables below to find the x-values where A. C. 0, 1, 2,6,7 B. 14, 29 D. 2,3 Earl s advice is to ignore the method they tell you to use and just solve the system. In fact, whenever you are told to use a particular method in a multiple choice test, treat it as a suggestion that you are free to ignore! Set the equations equal: Add 3 23: Result: Factor out a 3: Factor the trinomial: Solve for :, Answer D Page 31 of 42

32 42. Find the perimeter of the figure below: A C. 2 4 B D To find the perimeter, add the lengths of the four sides of the rectangle: Expression Side Bottom Left Top (same as bottom) Right (same as left) 4 82 Answer D 43. Which expression must be subtracted from 4 62 to result in 5 93? A C. 38 B. 35 D We want to find an expression such that: 4 62 expression 5 93 Adding expression to both sides and subtracting 5 93 from both sides of the equation gives something we can work with: expression Then, remembering to change the signs of 5 93 and add, we get: Answer B Page 32 of 42

33 44. Find the product: A. 15 C B D First, let s multiply two terms together, then multiply the result by the third term Then, multiply the result by Answer B 45. Multiply: A C B D Answer A 46. Factor completely: A. 9 4 C B D after rearranging terms. Answer D Page 33 of 42

34 47. Factor completely: A C B D To factor this expression, which is a difference of cubes, we must use the formula provided on the second page of this document: Let s proceed to solve this problem: Answer C 48. Solve: A. 1, 0, 5 3 C. 5 3,0,1 B. 3, 0, 5 D. 5, 0, 3 Alternatively, graph the function and see where it crosses the axis. Page 34 of 42

35 49. Solve: 3 10 A. 5,2 C. 5, 2 B. 5,2 D. 5, Solve: 360 A. 6, 6 C. 6,6 B. 6, 6 D. 6, One way to factor 72 is to rewrite the expression as 72. What is the equivalent of? A. 9 C. B. D. Page 35 of 42

36 52. What is the remainder in the division ? A. 250 C B. 650 D The easiest approach to this is to use synthetic division. First, note that the root implied by the divisor 5 is 5. Also, don t forget the 0 term in the first expression Answer A 53. Use synthetic or long division to find the quotient of ? A B. 2 1 C D. 2 1 The easiest approach to this is to use synthetic division. First, note that the root implied by the divisor 16 is The result is: Answer B For a full explanation of synthetic division, see pages of the Algebra Handbook or use the synthetic division section of the Algebra App, both of which are available at Page 36 of 42

37 54. Given 9 is a factor of the polynomial, , what are the remaining factors? A If 9 is a factor, then a root is: 9. B C. 4 9 D The result is: Recognize a difference of squares: 16x Factor the expression: Answer A 55. What is the end behavior for the function, 433 6? A. as, and as, B. as, and as, C. as, and as, D. as, and as, To determine end behavior, we need only know the lead term (the one with the highest exponent). To get this term, multiply the terms with the highest exponents in each of the factors given. Lead term of is: 3 3. Notice that the sign of the lead term is positive and that the exponent is odd. So, this functions will have end behavior the same as the function. That is: As, and as, Answer A Check out the following table that describes the end behavior of any polynomial: End Behavior of Polynomials Positive lead coefficient Acts like: Negative lead coefficient Acts like: Odd exponent on lead term Even exponent on lead term Page 37 of 42

38 56.. Note: We typically report relative maxima and minima using ordered pairs, which is preferred in higher level math classes. However, this problem is asking for the values only. How do we know this? Simply by looking at the form of the answers given. Page 38 of 42

39 58. Which polynomial is graphed below? A. 1 3 B C. 3 1 D. 3 1 This function appears to be a cubic equation with three real zeros at 1, 0, 3. Each zero,, corresponds to a factor of. The resulting equation has a factor for each zero, in particular: 1 3. Since the factors can appear in any order, we look through the answers to determine the correct answer has the three factors slightly rearranged: 31. Answer C 59. Which of the following is not a possible rational root of the function: ? 8 A. 1 C. 3 B. 4 D. 6 3 Possible roots are of the form where is a factor of the constant term, 8, and is a factor of the lead coefficient, 3. The factors are as follows: (factors of 8): 1, 2, 4, 8 (note: factors always have signs) (factors of 3): 1, 3 So, all possible rational roots have form: 1, 2, 4, 8,,,, Comparing these with the answers above, we see that answers A, B and C are in this list, but answer D, 6, is not. Answer D Page 39 of 42

40 60. Given 5, use synthetic division to find the two remaining real solutions of A. 2, 14 Note that one root is given to us directly: 5. 3 B. 6, C. 2, D. no other solutions The result is: Split the middle term: Pair terms: Factor pairs: Collect terms: 3x 20x 280 3x 6x 14x 280 3x 6x14x Break into separate equations: Solutions for in the equation:, Answer A 61. The equation is graphed below. Use the graph to help solve the equation and find all the roots of the function. A. 3,2,2 B. 12, 1, 3 C. 3, 2, 2 D. 12, 3 7, It is clear from the graph that one of the roots of the equation is 3. We could use synthetic division on the original equation to determine the quadratic that would remain when we extract that root. However, we can be smarter and save ourselves some work. It appears that the only real root of this equation occurs at 3. Only answer C to this question that has a single real root equal to 3. Answers A and B have more than one real root, and answer D does not have 3 as a root. Answer C Page 40 of 42

41 62. A manufacturer is going to package their product in an open rectangular box made from a single flat piece of cardboard. The box will be created by cutting a square out from each corner of the rectangle and folding the flaps up to create a box. The original rectangular piece of cardboard is 20 long and 15 wide. Write a function that represents the volume of the box A C B D The length of the box is: The width of the box is: The height of the box is: Volume of the box is: Answer B Page 41 of 42

42 63. Sketch the graphs of and on the same coordinate plane given the following information: has zeros at 6, 1, 2 As, and as, has a local minimum at 1, 5 and a local maximum at 4, 9 5 How many real solutions exist when? A. B. 1 C. 2 D. 3 Let s see what each item tells us about the curves: has zeros at 6, 1, 2. This suggests that is a cubic (it has 3 zeros). As, and as,. This tells us that the degree of the curve is odd (this is consistent with the suggestion that it is a cubic) and that its lead coefficient is positive. So it looks like this: has a local minimum at 1, 5 and a local maximum at 4, 9. This is consistent with our shape; the minimum is to the right of the maximum. 5. This is a separate function, a horizontal line at 5. This problem is asking us how many places the two functions intersect. A sketch of the two curves is shown below. Notice that the line 5, being between the minimum 5 and maximum 9 of, must intersect in three places if is a cubic. Answer D Note: as described, could be 5 th, 7 th, or any other odd degree polynomial, in which case could have more than three real solutions (points of intersection). Therefore, a better answer to this question would be at least 3 real solutions. Page 42 of 42